No, there isn't storage for things like that and some attractions you will be exiting at a different place than where you got on, unless you are boarding where wheelchair users board.
In the past, some people have used canes that have a little seat that folds down from the cane and makes a seat. The last post I remember reading about those was someone posting about a year ago saying that they were not allowed to bring their cane/stool in because it was considered a hazard. I haven't seen anyone post after that to say whether they were allowed or not.
I know that actual chairs are not allowed.
If your nephew is concerned about getting tired, he may want to consider a wheelchair or an
ECV (motorized mobility
scooter - only rented for use by people over 18). The distances involved in a WDW trip are immense - it's not unusual to walk over 5 miles in a day! There is more information about wheelchairs and
ECVs in the disABILITIES FAQs thread.